Momentum maps and Noether theorem for generalized Nambu mechanics

On symmetries and conserved quantities in Nambu mechanics

In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number. The origin of the difference may be traced back to a shift in degrees of relevant forms present in equations of motion, or, alternatively, to a corresponding shift in degrees of relevant objects in action integral for Nambu mechanics.

Vector Hamiltonians in Nambu mechanics

We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in $\mathbb{R}^n$ can be represented as a generalized Nambu mechanics with $n-1$ integral invariants. For the case when the phase flow in $\mathbb{R}^n$ has $n-3$ or less first integrals, we introduce the Cartan concept of mechanics. As an example we give the fifth integral invariant of Euler top.

欧拉-庞加莱约减与含输运参数和附加动力学的离散机械系统的凯尔文-诺特定理

Euler--Poincaré reduction and the Kelvin--Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

本文提出离散欧拉-庞加莱约减方法，用于具有输运参数和附加动力学的离散拉格朗日系统，通过群差分映射技术扩展了连续和离散的凯尔文-诺特定理，并应用于水下机器人动力学的数值模拟。

Noether对称群、局部守恒积分与经典力学中的动力学对称性

Noether symmetry groups, locally conserved integrals, and dynamical symmetries in classical mechanics

本文通过混合拉格朗日-哈密顿框架，探讨守恒积分与对称性的关系，通过三个例子展示局部Liouville可积性的推广，利用变分对称性得到局部守恒积分，并引入动作-角度变量实现局部时间内的显式积分。

Revisiting Noether's Theorem on constants of motion

In this paper we revisit Noether's theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By recognizing that the Bessel-Hagen (BH) function need not be a total time derivative, we can easily deduce nonlocal constants of motion. We prove that we can always trivialize either the time change or the BH-function, so that, in particular, BH-invariance turns out not to be more general than Noet…